The boring answer is, "sometime after 1840," which is around when the term "scientist" was coined by William Whewell. If one does an Ngram search for "scientist" one finds that it is not really until the late 19th century that the term takes off.

The more interesting answer is more like: science was we know it essentially "professionalized" in the early-to-mid 19th century. What we mean by "professionalized" is that it went from something that was done by the wealthy or highly-eccentric and became a regular sort of "profession" with standard career trajectories, institutions of support, and a sense of itself as something one "could do." This is not to suggest the old way changed overnight (and several important late-19th century scientists, like Darwin and Galton and many others, were essentially pre-professional in many ways), but it does reflect the changing trend in Europe (and a little bit later, globally).

It is of note that "scientist" was not the only possible name for the profession. "Man of science," "and "scientific worker" were preferred by many writers. In 1880, the head of the AAAS made the following remark on the term "scientist" and why he didn't like it:

The word “scientist” is a coinage of the newspaper reporter, and, as ordinarily used, is very comprehensive. Webster defines a scientist as being “one learned in science, a savant,”—that is, a wise man,—and the word is often used in this sense. But the suggestion which the word conveys to my mind is rather that of one whom the public suppose to be a wise man, whether he is so or not, of one who claims to be scientific.

It is separately of note that, from what I can tell, it was rare at the time for someone to specifically be called a "natural philosopher" — more commonly they were just called "philosophers" without the distinction of the natural being added. It is interesting to me that it is often more important to us today to add the "natural" than it seems to have been in the past — because we want to simultaneously emphasize that it was considered a form of philosophy, but we also feel the need to impose our present-day distinctions about the differences between "science" and "philosophy" backwards, I suspect.

For an excellent discussion of the professionalization of science, see Steven Shapin, The Scientific Life: A Moral History of a Late Modern Vocation (University of Chicago Press, 2008). Chapter 1 in particular covers several interesting aspects about why the idea of a "scientist" was resisted by some: it seemed to make individual a procedure that, ideally, should be collective and not identified with any particular person. For the AAAS quote, see p. 322, fn. 7.